PHYSICAL REVIEW B 73, 212302 共2006兲

Vibrational modes in aperiodic one-dimensional harmonic chains F. A. B. F. de Moura Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió, AL, Brazil

L. P. Viana and A. C. Frery Instituto de Computação, Universidade Federal de Alagoas, 57072-970 Maceió, AL, Brazil 共Received 2 December 2005; revised manuscript received 21 March 2006; published 6 June 2006兲 The present paper addresses the effect of aperiodicity in one-dimensional oscillatory systems. We study the nature of collective excitations in harmonic chains in the presence of aperiodic and pseudorandom mass distributions. Using the transfer matrix method and exact diagonalization on finite chains, we compute the localization length and the participation number of eigenmodes within the band of allowed frequencies. Our numerical calculations indicate that, for aperiodic arrays of masses, a new phase of extended states appears in this model. For pseudorandom masses distribution, all eigenstates remain localized except the uniform mode 共␻ = 0兲. Solving numerically the Hamilton equations for momentum and displacement of the chain, we compute the spreading of an initially localized energy excitation. We show that, independent of the kind of initial excitation, an aperiodic structure of masses can induce ballistic transport of energy. DOI: 10.1103/PhysRevB.73.212302

PACS number共s兲: 63.50.⫹x, 63.22.⫹m, 62.30.⫹d

INTRODUCTION

The simplest and almost unique class of systems for which one can perform analytic calculations is represented by harmonic chains. Even though they are characterized by a peculiar dynamics, basically because of the integrability of the motion, their behavior can help to shed some light on various aspects of heat conductivity. One of the properties of harmonic chains is the possibility to decompose the heat flux into the sum of independent contributions associated with the various eigenmodes.1 Particularly, the heat flux in lowdimensional classical systems has been the target of recent intense investigations.2–13 The thermal conductivity of harmonic and anharmonic chains with uncorrelated random masses,4 as well as that of a chain of hard-point particles with alternate masses,5 have been numerically investigated in detail. The main issue here is whether these systems display finite thermal conductivity in the thermodynamic limit, a question that remains controversial.6 In general, the behavior of the thermal conductivity, as well as the vibrational eigenmodes, appears to be determined by the disorder and anharmonicity.14 When aperiodic and disordered structures are involved, the Anderson theory plays a key role in the localization properties of vibrational eigenfunctions in condensed matter physics.15,16 The localization of collective excitation by a random potential is a quite general feature. It applies, for example, to the study of magnon localization in random ferromagnets.17 Further, the collective vibrational motion of one-dimensional 共1D兲 disordered harmonic chains of N random masses can also be mapped onto a one-electron tight-binding model.18 In such a case, most of the normal vibrational modes are localized. However, there are a few low-frequency modes not localized, whose number is of the order of 冑N.18,19 It was shown that correlations in the mass distribution produce a new set of nonscattered modes in this system.20 Nonscattered modes have also been found in disordered harmonic chains with dimeric correlations in the spring constants.21 A large amount of work has 1098-0121/2006/73共21兲/212302共4兲

been done in recent decades to understand localization behavior in randomly disordered chains.12 Most of this work has been concentrated in uncorrelated,15 short-range,22 and long-range23 correlated disorder. Among these models, chains with diluted disorder, another kind of short-range correlation, have attracted renewed interest.24–29 In general, the model consists of two interpenetrating sub-lattices, one composed of random potentials 共Anderson lattice兲 and the other composed of nonrandom sites of constant potential. Due to the periodicity, special resonant energies appear. Recently, the effect of this kind of local correlations in a disordered harmonic chain was studied.29 There is another class of 1D models, the aperiodic Anderson model,33 lying between the random Anderson model and the periodic Bloch model. They have been extensively investigated in the literature30–34 and the localized or extended nature of their eigenstates has been related to general characteristics of the aperiodic on-site distributions. Within the line of vibration modes in classical chains, the roles played by aperiodic structure onto localizations properties and/or energy transport have not been completely studied. A very interesting study was publishied in Ref 11. The authors studied the effect of a Fibonacci-like array of masses on the thermal conductivity of a onedimensional anharmonic lattice. In this work, we focus on the effect of aperiodicity in one-dimensional harmonic systems. Here we will consider the mass distributions following the sequence used in Ref. 33. From this we can simulate both aperiodic and pseudorandom mass distributions. We use the transfer matrix method to obtain accurate estimates for the Lyapunov exponent. These results are used to characterize the nature of vibrational modes in this model. We show that, due to the aperiodicity of the mass array, the low-frequency extended vibrational modes can exist. All calculations were confirmed using an exact diagonalization procedure on a finite chain. In order to study the time evolution of an initially localized energy input, we calculate the second moment M 2共t兲 of the energy spatial distribution. We show that, independent of the kind of

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BRIEF REPORTS

FIG. 1. 共a兲 Lyapunov coefficient ␥ vs ␻2 for ␯ = 0.2 共䊐兲, 0.4 共䉮兲, 0.6 共⌬兲, and 0.8 共䉯兲; 共b兲 the same for ␯ = 1.4 共〫兲 and 1.6 共䉰兲. 共c兲 Scaled participations number ␰ / N for ␯ = 0.4 and 0.6 and distinct system size. 共d兲 Scaled participations number ␰ / N for ␯ = 1.4 and 1.6 and distinct system size. All previous calculations were done for W = 2. These results indicate that, for ␯ ⬍ 1, there are extended vibrational modes at the lowfrequency region. 共e兲 Lyapunov coefficient ␥ vs ␻2 for ␯ = 0.4, 0.6 and W = 1, 2, and 3.8. 共f兲 The scale function for dates of 共c兲. The critical frequency that separates extended from localized vibrational modes is independent of the ␯ exponent; it depends only on the width W of mass distributions, see 共e兲. We use m0 = 4 in all cases and W = 2 in 共a兲–共d兲 and 共f兲.

initial excitation, an aperiodic structure of masses can induce a ballistic energy transport. Therefore, in the thermodynamic limit the model presents extended states. This modifies the heat conduction of the harmonic chain. VIBRATIONAL MODES

We start by considering a disordered harmonic chain of N masses, for which the equation of motion for the displacements qn = unexp共i␻兲 with vibrational frequency ␻ is 19,20 共␤n−1 + ␤n − ␻2mn兲un = ␤n−1un−1 + ␤nun+1 .

共1兲

Here, we shall consider a mass distribution given by mn = m0 + W cos共␣n␯兲,

兩QNc共0兲兩 1 log , 兩c共0兲兩 N→⬁ N

␥ = lim

共3兲

where c共0兲 = 共 u10 兲 is a generic initial condition and QN is the product of all transfer matrices, u

N

n=1

冉

2 − m n␻ 2 − 1 1

0

2

冊

.

共4兲

In addition, we compute the participation ratio P defined by N N u2n共f 2兲 / 兺n=1 u4n共f 2兲,35 where the Fourier un共f 2兲 are P共f 2兲 = 兺n=1

2

2

+⌬␻ 2 2 = N1f 兺 ff 2==␻␻2−⌬ ␻2 P共f 兲, where ⌬␻ = 0.05 and N f is the number of eigenmodes within each interval 关␻2 − ⌬␻2 , ␻2 + ⌬␻2兴. A more quantitative scaling analysis of the participation number trend can be derived by introducing the set of auxiliary functions

冋冏

共2兲

with ␣ an arbitrary rational number and ␯ and W being a tunable parameter.33 We use m0 ⬎ W to avoid negative masses. In our calculations, we will use units where all elastic force couplings ␤n are equivalent and equal to 1. The localization length of each vibrational mode is taken as the inverse of the Lyapunov exponent ␥ defined by19,20

QN = 兿

those associated with an eigenmode f 2 of a chain of N masses and are obtained by direct diagonalization of the N ⫻ N secular matrix A defined by Ai,i = 2 / mi, Ai,i+1 = Ai+1,i = 1 / 共mimi+1兲1/2, and all other Ai,j = 0.35 P共f 2兲 displays a dependence on the chain size for extended states and is finite for exponentially localized ones. In our calculations, we compute the average participation number defined by ␰共␻2兲

⌰共␻2,N1,N2兲 = exp −

N2 N1 − ␰共␻2,N2兲 ␰共␻2,N1兲

冏册

,

共5兲

which is a measure of the difference between data from two consecutive chain sizes investigated. For extended states, ⌰ ⬇ 1 for large chain sizes. For localized states, ⌰ → 0 in the thermodynamic limit. In Fig. 1共a兲, we show the Lyapunov exponent ␥ as a function of ␻2 obtained from the transfer matrix method for ␯ = 0.2 共䊐兲, 0.4 共䉮兲, 0.6 共⌬兲, and 0.8 共䉯兲. Here we use m0 = 4 in all cases and W = 2 in Figs. 1共a兲–1共d兲 and 1共f兲. One can see that this exponent vanishes in the low-frequency region 关␻2 ⬍ ␻2c ⬇ 0.6共1兲兴. This feature is a clean signature of extended vibrational modes. In Fig. 1共b兲, we show the Lyapunov exponent ␥ as a function of ␻2 for ␯ = 1.4 共〫兲, and 1.6 共䉰兲. As this coefficient does not vanish for ␻2 ⬎ 0, for ␯ ⱖ 1 all eigenstates are localized for ␻ ⬎ 0. In Fig. 1共c兲, we show data for the scaled participation number for ␯ = 0.4, 0.6 and N = 1000, 2000, and 8000. The well-defined

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data collapse in the low-frequency region confirms the extended nature of these vibrational modes. For ␯ ⬎ 1 关see Fig. 1共d兲兴, only for ␻ = 0 are the participation number scales proportional to the system size. The critical frequency ␻2c that appears in Figs. 1共a兲–1共c兲 separating extended from localized vibrational modes does not depend on the ␯ exponent. In Fig. 1共e兲, one can see that it depends only on the width W of the mass distributions. Both results for participation number 关Figs. 1共c兲 and 1共d兲兴 are in perfect agreement with the calculated Lyapunov exponents 关Figs. 1共a兲 and 1共b兲兴. In Fig. 1共f兲, the ⌰ versus ␻2 data suggest that the phase of extended low-frequency vibrational modes is stable in the thermodynamic limit and that the critical frequency ␻2c does not depend on the ␯ exponent for ␯ ⬍ 1. We use distinct system sizes 共N = 5 ⫻ 106, 107, 2 ⫻ 107, and 4 ⫻ 107兲 to verify if our results are due to a finite-size effect. For all studied system sizes, we obtain a linear vanishing exponent 共␥ ⬀ 1 / N, not shown here兲 that signs for true extended states in the thermodynamics limit 共N → ⬁ 兲. This behavior does not completely assure the existence of extended states, as in the case of a vibrational wave envelope displaying a power-law decaying,15,19 thus we further study the dynamics of an initially localized excitation in the chain to understand energy spread in this system.

ENERGY TRANSPORT

In order to study the time evolution of a localized energy pulse, we calculate the second moment of the energy distribution.3,36 This quantity is related to the thermal conductivity by Kubo’s formula.3 The classical Hamiltonian H for a harmonic chain can be written as

FIG. 2. The time-scaled second moment M 2共t兲 / ta vs time t obtained by using initial impulse excitation. 共a兲 For m0 = 4, W = 2, and ␯ = 0.2, 0.4, and 0.6, the system displays ballistic spreading 关M 2共t兲 ⬀ t2兴. 共b兲 The same for ␯ = 1.5, 2.0, and 2.5; in this case the system is in a superdiffusive regime 关M 2共t兲 ⬀ t1.5兴. For ␯ ⬍ 1.0, the frequencies below ␻c are extended over the lattice. This causes the ballistic energy spread of the pulse.

In Figs. 2 and 3 we plot, for impulse and displacement initial excitations, respectively, the time-scaled second moment M 2共t兲 / ta as a function of time for a 1D aperiodic harmonic chain with ␯ ⬍ 1.0 共a兲 and ␯ ⬎ 1.0 共b兲, using N = 1.6 ⫻ 104. For ␯ ⬍ 1.0, the extended modes for frequencies below ␻c induce ballistic energy spread of the pulse, i.e., a = 2 independent of the kind of initial excitation. For ␯ ⬎ 1.0, the pseudorandom character of the mass chain induces nonscattered 共extended兲 modes only for frequencies very close to zero. Therefore, we obtain a dynamic behavior similar to those found on the uncorrelated random harmonic chain: a = 1.5 for impulse initial excitation and a = 0.5 for displacement initial excitation.3

N

H = 兺 hn共t兲,

共6兲

n=1

where the energy hn共t兲 at site n is given by hn共t兲 =

P2n 1 + 关共Qn+1 − Qn兲2 + 共Qn − Qn−1兲2兴. 2mn 4

共7兲

SUMMARY AND CONCLUSIONS

We study the nature of collective excitations in harmonic chains with aperiodic and pseudorandom mass distributions given by Eq. 共2兲. Using a transfer matrix method and exact diagonalization on finite chains, we compute the localization length and the participation number of eigenmodes within the band of allowed frequencies. We observe that, for ␯ ⬍ 1,

Here Pn and Qn define the momentum and displacement of the nth site mass. The fraction of the total energy H at site n is given by hn共t兲 / H and the second moment of the energy distribution, M 2共t兲, is defined by3 N

M 2共t兲 = 兺 关n − 具n共t兲典兴2关hn共t兲/H兴,

共8兲

n=1

N where 具n共t兲典 = 兺n=1 n关hn共t兲 / H兴. An initial excitation is introduced at the site n0 at t = 0. Using the fourth-order Runge-Kutta method, we solve the Hamilton equations for Pn共t兲 and Qn共t兲 to calculate M 2共t兲. In harmonic chains, the energy spreads faster for an initial impulse excitation than for an initial displacement excitation,3,36 in such a way that the behavior is strongly dependent on the initial conditions. We have performed calculations for both kinds of initial excitation.

FIG. 3. The time-scaled second moment M 2共t兲 / ta vs time t obtained by using initial displacement excitation. 共a兲 For m0 = 4, W = 2, and ␯ = 0.2, 0.4, and 0.6, as in Fig. 2共a兲, ballistic energy spreading is observed 共a = 2兲 associated with the low-frequency extended vibrational eigenstates. 共b兲 For ␯ = 1.5, 2.0, and 2.5 the pseudorandom character of random masses induces a subdiffusive behavior 共a = 0.5兲.

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the Lyapunov exponent vanishes and the participation number diverges with N in the low-frequency region; therefore, there is a new phase of extended vibrational modes in these aperiodic-mass chains. We also show that the presence of these nonscattered vibrational modes can modify the energy spreading of an initially localized excitation. By calculating the second moment M 2共t兲 of the energy spatial distribution, we find that, associated with the emergence of a phase of low-energy extended collective excitations, M 2共t兲 displays ballistic propagation of the energy pulse independent of the kind of initial excitation. For ␯ ⬎ 1, the pseudorandom character of mass array induces a similar behavior to those found in harmonic chains with uncorrelated random mass distributions, sub- or superdiffusive depending on the kind of initial excitation, impulse or displacement-like, respectively.3 The authors of Ref. 11 found that the thermal conductivity at low temperatures of a one-dimensional nonlinear lattice is strongly dependent on the nature of the harmonic eigenstates. They found constant flux for a periodic lattice and normal

heat flux for disordered system; this scenario happens due to the extended and localized behavior of the eigenstates, respectively. For the aperiodic kind of system that they studied, the heat flux appears to be power law divergent, and the strength of divergence is strongly related to the specific structure of the model and, clearly, to the degree of localizations of their eigenstates. In light of these previous results, the presence of a finite phase of extended harmonic modes at low frequencies and the anomalous energy spreading found in our model also suggests an abnormal thermal conductivity in agreement with Ref. 11. We expect that the present work will stimulate further studies along this direction.

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ACKNOWLEDGMENTS

This work was partially supported by the Brazilian research agency CNPq and by the Alagoas state research agency FAPEAL.

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